Braided fusion categories encode data from such diverse areas of
mathematics as quantum computation, representation theory, topological
quantum field theory, knot theory, von Neumann algebras, and other
assorted topics in mathematical physics. We will discuss the extent to
which these categories have been classified, a variety of tools which
have been constructed for this purpose, and optimistically conjecture
about progress in the near future.
If you have studied the representation theory of finite groups, then you
will be able to relate to a good portion of this talk.